Divisibility, Factors, GCF, Multiples, LCM, Prime, Composite, &amp; Prime Factorization

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Divisibility, Factors, GCF, Multiples, LCM, Prime, Composite, &amp; Prime Factorization. Divisibility. Divisibility – a number is divisible by another number when you divide them and there is NOT a remainder (or decimal) Calculator Time!!!!!. Divisibility con…. Example: 35 ÷ 5 = 7

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### Divisibility, Factors,GCF, Multiples, LCM, Prime, Composite, & Prime Factorization

Divisibility

Divisibility – a number is divisible by another number when you divide them and there is NOT a remainder (or decimal)

Calculator Time!!!!!

Divisibility con…

Example: 35 ÷ 5 = 7

So…

35 is divisible by 5 because

there is no remainder/decimal

Example: 42 ÷ 5 = 8.4

So…

42 is NOT divisible by 5 because

there isa remainder/decimal

X

Divisibility con…

Is 45 divisible by 2, 3, 4, 5, 6, 8, 9, or 10?

45 ÷ 2 = 22.5

45 ÷ 3 = 15

45 ÷ 4 = 11.25

45 ÷ 5 = 9

45 ÷ 6 = 7.5

45 ÷ 8 = 5.625

45 ÷ 9 = 5

45 ÷ 10 = 4.5

X

So…..

45 is divisible by:

3, 5, and 9

X

X

X

X

Divisibility con…

Is 60 divisible by 2, 3, 4, 5, 6, 8, 9, or 10?

60 ÷ 2 = 30

60 ÷ 3 = 20

60 ÷ 4 = 15

60 ÷ 5 = 12

60 ÷ 6 = 10

60 ÷ 8 = 7.5

60 ÷ 9 = 6.67

60 ÷ 10 = 6

So…..

60 is divisible by:

2, 3, 4, 5, 6, and 10

X

X

Factors

Factors – the numbers we multiply together to get a product

8 x 4 = 32

Factor

Factor

Product

Factors con…

There is a specific set of factors for each number.

To find factors:

- take number and divide it by 1, then 2, then 3, then 4, etc.

- if you get a decimal – it is not a factor

- no decimal – the factor pair is the # you divided by and the answer

Factors con…

List all the factors of 48:

48 ÷ 1 = 48

48 ÷ 2 = 24

48 ÷ 3 = 16

48 ÷ 4 = 12

48 ÷ 5 = 9.6

48 ÷ 6 = 8

48 ÷ 7 = 6.86

Factors:

so

= 48

1 x 48

= 48

so

2 x 24

= 48

so

3 x 16

= 48

so

4 x 12

no

so

6 x 8

= 48

no

Factors con…

List all the factors of 36:

36 ÷ 1 = 36

36 ÷ 2 = 18

36 ÷ 3 = 12

36 ÷ 4 = 9

36 ÷ 5 = 7.2

36 ÷ 6 = 6

Factors:

so

= 36

1 x 36

= 36

so

2 x 18

= 36

so

3 x 12

= 36

so

4 x 9

no

so

6 x 6

= 36

Factors con…

List all the factors of 45:

45 ÷ 1 = 45

45 ÷ 2 = 22.5

45 ÷ 3 = 15

45 ÷ 4 = 11.25

45 ÷ 5 = 9

45 ÷ 6 = 7.5

Factors:

so

1 x 45

= 45

no

so

3 x 15

= 45

no

so

5 x 9

= 45

no

Greatest Common Factor (GCF)

Biggest Factor they have in common

Find the GCF of 36 and 24:

3624

GCF = 12

1 x 36

1 x 24

2 x 12

2 x 18

3 x 8

3 x 12

4 x 9

4 x 6

6 x 6

GCF con…

Find the GCF of 60 and 90:

6090

GCF = 30

1 x 60

1 x 90

2 x 30

2 x 45

3 x 20

3 x 30

4 x 15

5 x 18

6 x 15

5 x 12

9 x 10

6 x 10

GCF con…

Find the GCF of 45 and 72:

4572

GCF = 9

1 x 45

1 x 72

3 x 15

2 x 36

5 x 9

3 x 24

4 x 18

6 x 12

8 x 9

GCF con…

Find the GCF of 30, 45, and 70:

304570

1 x 30

1 x 45

1 x 70

2 x 15

3 x 15

2 x 35

3 x 10

5 x 9

5 x 14

5 x 6

7 x 10

GCF = 5

### Prime, Composite, andPrime Factorization

• Examples:
• Special Case: the number 2 is the only even number that is prime!!!

2

3

5

7

11

1 x 2

1 x 3

1 x 5

1 x 7

1 x 11

Composite Numbers – have more than 2 factors

• Examples:

4

24

12

9

1 x 4

1 x 24

1 x 12

1 x 9

2 x 2

2 x 12

2 x 6

3 x 3

3 x 8

3 x 4

1,3,9

1,2,4

4 x 6

1,2,3,4,6,12

1,2,3,4,6,8,12,24

All even numbers except the #2 are composite

• Why?
• All even numbers are divisible by 2
• so 2 is going to be a factor
• - Is the # 1 prime or composite?

1

The number 1 is neither prime nor composite – it has only one factor – the number 1

1 x 1

Practice: Tell whether each number is prime or composite

25

54

19

32

23

1 x 19

1 x 32

1 x 23

1 x 25

1 x 54

Prime

2 x 16

Prime

5 x 5

2 x 27

3 x 18

4 x 8

Composite

6 x 9

Composite

Composite

Prime Factorization – use a factor tree to factor until all that is left is prime numbers

32

Prime Factorization =

2 x 2 x 2 x 2 x 2

or

25

4

8

2

2

2

4

2

2

45

Prime Factorization =

3 x 3 x 5

or

32 x 5

5

9

3

3

36

Prime Factorization =

2 x 2 x 3 x 3

or

22 x 32

6

6

2

3

2

3

Multiples – is the product when you multiply

• There are no set amount – there is an infinite number of multiples
• The multiples are equal to or larger than the starting number
• Example:
• Multiples of 5: 5, 10, 15, 20, 25, 30…….

How do you find multiples?

• Take the numberandmultiply it by another whole number – I like to go in orderstarting with the number 1
• It’s like we are counting by that number

Examples:

List five multiples of the # 8

1 x 8 = 8

2 x 8 = 16

3 x 8 = 24

4 x 8 = 32

5 x 8 = 40

8, 16, 24, 32, and 40 are multiples of the number 8

Examples:

List five multiples of the # 6

1 x 6 = 6

2 x 6 = 12

3 x 6 = 18

4 x 6 = 24

5 x 6 = 30

6, 12, 18, 24, and 30 are multiples of the number 6

Examples:

List five multiples of the # 12

1 x 12 = 12

2 x 12 = 24

3 x 12 = 36

4 x 12 = 48

5 x 12 = 60

12, 24, 36, 48, and 60 are multiples of the number 12

Least Common Multiple (LCM)– smallest multiple a set of numbers have in common

• How to find LCM:
• 1.) Begin with the larger number
• 2.) List 5 – 10 multiples (go in order)
• 3.) Begin listing multiples of the other #
• 4.) Stop when you find a multiple they have in common
• 5.) If needed, go back and list more multiples of the first number

Example: Find the LCM of 6 and 7

Multiples of 7:

7,

14,

21,

28,

35,

42,

49,

56,

63,

70

Multiples of 6:

12,

36,

6,

18,

24,

30,

42

LCM of 6 & 7 = 42

Example: Find the LCM of 15 and 20

Multiples of 15:

15,

30,

45,

60,

75,

90

Multiples of 20:

20,

40,

60

LCM of 15 & 20 = 60

Example: Find the LCM of 5, 6, and 15

Multiples of 15:

15,

30,

45,

60,

75

Multiples of 6:

12,

6,

18,

24,

30

Multiples of 5:

10,

5,

15,

20,

25,

30

LCM of 5, 6, & 15 = 30

LCM in Word Problems:

On every third page of Sarah’s scrapbook, she has a friend’s signature. On every fourth page she has a teacher’s signature. What is the page number of the first page that will have both signatures?

How to work: find the LCM of 3 and 4

Multiples of 4:

8,

24,

4,

12,

16,

20,

28

Multiples of 3:

LCM of 3 & 4 = 12

6,

3,

9,

12,

LCM in Word Problems:

Scrapbook stickers come in packages of 10, and labels come in packages of 25. What is the least number of packages of each Sarah should buy if she needs to have an equal number of stickers and labels?

Multiples of 25:

Stickers – 5 packs

Labels – 2 packs

50,

25,

75,

100,

125

1

2

Multiples of 10:

20,

40,

10,

30,

50

1

2

3

4

5